Thus, the lines representing the given pair of equations are parallel to each other and hence, these lines will never intersect each other at any point or there is no possible solution for the given pair of equations.

3. On comparing the ratios a1/a2, b1/b2 and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5

2x – 3y = 7

(ii) 2x – 3y = 8

4x – 6y = 9

(iii) 3/2 x + 5/3 y = 7

9x – 10y = 14

(iv) 5x – 3y = 11

-10x + 6y = -22

(v) 4/3 x + 2y = 8

2x + 3y = 12

Solution:

(i) 3x + 2y = 5

2x – 3y = 7

a1/a2 = 3/2

b1/b2 = -2/3

c1/c2 = 5/7

a1/a2 ≠ b1/b2

These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(ii)2x − 3y = 8

4x − 6y = 9

a1/a2 = 2/4 = 1/2

b1/b2 = -3/-6 = 1/2

c1/c2 = 8/9

a1/a2 = b1/b2 ≠ c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 3/2 x + 5/3 y = 7

9x – 10y = 14

a1/a2 = (3/2)/9 = 3/18 = 1/6

b1/b2 = (5/3)x(-10) = -1/6

c1/c2 = 7/14 = 1/2

a1/a2 ≠ b1/b2

These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(iv) 5x – 3y = 11

-10x + 6y = -22

a1/a2 = 5/-10 = -1/­2

b1/b2 = -3/6 = -1/2

c1/c2 = 11/-22 = -1/2

a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

(v) 4/3 x + 2y = 8

2x + 3y = 12

a1/a2 = (4/3)/2 = 2/­3

b1/b2 = 2/3

c1/c2 = 8/12 = 2/3

a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

4. Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically:

(1) x + y = 5

2x + 2y = 10

(2) x – y = 8

3x – 3y = 16

(3) 2x + y – 6 = 0

4x – 2y – 4 = 0

(4) 2x – 2y – 2 = 0

4x – 4y – 5 = 0

Solution:

(1) x + y = 5

2x + 2y = 10

a1/a2 = 1/2

b1/b2 = 1/2

c1/c2 = 5/10 = 1/2

So, we have, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

x + y = 5,

x = 5 – y

x

4

3

2

y

1

2

3

and 2x + 2y = 10

x = (10-2y)/2

x

4

3

2

y

1

2

3

the graphic representation is,

From the graph, we can observe that these lines are overlapping each other. Therefore, infinite solutions are possible for the given pair of equations.

(2) x – y = 8

3x – 3y = 16

a1/a2 = 1/3

b1/b2 = -1/-3 = 1/3

c1/c2 = 8/16 = 1/2

So, we have, a1/a2 = b1/b2 ≠ c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent

(3) 2x + y – 6 = 0

4x – 2y – 4 = 0

a1/a2 = 2/4 = 1/2

b1/b2 = -1/2

c1/c2 = -6/-4 = 3/2

So, we have, a1/a2 ≠ b1/b2

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

2x + y − 6 = 0,

y = 6 − 2x

x

0

1

2

y

6

4

2

and 4x − 2y − 4 = 0 y = (4x-4)/2

x

1

2

3

y

0

2

4

the graphic representation is,

From the graph we can observe that these lines are intersecting each other at the only point i.e., (2, 2) and it is the solution for the given pair of equations.

(iv) 2x – 2y – 2 = 0

4x – 4y – 5 = 0

a1/a2 = 2/4 = 1/2

b1/b2 = -2/-4 = 1/2

c1/c2 = 2/5 = 2/5

So, we have, a1/a2 = b1/b2 ≠ c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent

5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Solution:

Let the width of the garden be x and length be y.

By data given in the problem

y − x = 4 ………………. (1)

y + x = 36 ……………..(2)

y − x = 4

x

0

8

12

y

4

12

16

y + x = 36

x

0

36

16

y

36

0

20

the graphic representation is,

From the graph we can observe that these lines are intersecting each other at only point i.e., (16, 20). Therefore, the length and width of the given garden is 20 m and 16 m respectively.

5. Given the linear equation 2x + 3y − 8 = 0, write another linear equations in two variables such that the geometrical representation of the pair so formed is:

(i) Intersecting lines

(ii) Parallel lines

(iii) Coincident lines

Solution:

(i) Intersecting lines

For this,

a1/a2 ≠ b1/b2

The second line such that it is intersecting the given line is

2x + 4y – 6 = 0 as a1/a2 = 2/2 = 1 and b1/b2 = 3/4 ; a1/a2 ≠ b1/b2

(ii) Parallel lines:

For this condition,

a1/a2 = b1/b2≠ c1/c2

Hence the second line can be

4x + 6y – 8 = 0

as a1/a2 = 2/4 = 1/2 ; b1/b2 = 3/6 =1/2 ; c1/c2 = -8/-8 = 1

Clearly, a1/a2 = b1/b2 ≠ c1/c2

(iii) coincident lines

For this,

a1/a2 = b1/b2 = c1/c2

Hence 6x + 9y – 24 = 0 be the second line

a1/a2 = 2/6 = 1/3; b1/b2 = 3/9 = 1/3 ; c1/c2 = -8/-24 = 1/3

Thus a1/a2 = b1/b2 = c1/c2

Draw the graphs of the equations x − y + 1 = 0 and 3x + 2y − 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Solution:

x − y + 1 = 0

x = y – 1

x

0

1

2

y

1

2

3

3x + 2y – 12 = 0

x = (12-2y)/3

x

4

2

0

y

0

3

6

the graphic representation is

From the figure, it can be observed that these lines are intersecting each other at point (2, 3) and x-axis at (−1, 0) and (4, 0).

Therefore, the vertices of the triangle are (2, 3), (−1, 0), and (4, 0).

It’s shocking how I didn’t get very far past the first few problems when I was already lost. I mean, I was an English major but still. Struggling through to the end makes me feel a bit better than I was yesterday though so thank you.